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Chapter 4 Discounted Cash Flow Valuation McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Key Concepts and Skills Be able to compute the future value and/or present value of a single cash flow or series of cash flows Be able to compute the return on an investment Be able to use a financial calculator and/or spreadsheet to solve time value problems Understand perpetuities and annuities McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter Outline 4.1 Valuation: The One-Period Case 4.2 The Multiperiod Case 4.3 Compounding Periods 4.4 Simplifications 4.5 What Is a Firm Worth? McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. 4.1 The One-Period Case If you were to invest $10,000 at 5-percent interest for one year, your investment would grow to $10,500. $500 would be interest ($10,000 × .05) $10,000 is the principal repayment ($10,000 × 1) $10,500 is the total due. It can be calculated as: $10,500 = $10,000×(1.05) The total amount due at the end of the investment is call the Future Value (FV). McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Future Value In the one-period case, the formula for FV can be written as: FV = C0×(1 + r)T Where C0 is cash flow today (time zero), and r is the appropriate interest rate. McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Present Value If you were to be promised $10,000 due in one year when interest rates are 5-percent, your investment would be worth $9,523.81 in today’s dollars. $9,523.81 $10,000 1.05 The amount that a borrower would need to set aside today to be able to meet the promised payment of $10,000 in one year is called the Present Value (PV). Note that $10,000 = $9,523.81×(1.05). McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Present Value In the one-period case, the formula for PV can be written as: PV C1 1 r Where C1 is cash flow at date 1, and r is the appropriate interest rate. McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Net Present Value The Net Present Value (NPV) of an investment is the present value of the expected cash flows, less the cost of the investment. Suppose an investment that promises to pay $10,000 in one year is offered for sale for $9,500. Your interest rate is 5%. Should you buy? McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Net Present Value NPV $9,500 $10,000 1.05 NPV $9,500 $9,523.81 NPV $23.81 The present value of the cash inflow is greater than the cost. In other words, the Net Present Value is positive, so the investment should be purchased. McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Net Present Value In the one-period case, the formula for NPV can be written as: NPV = –Cost + PV If we had not undertaken the positive NPV project considered on the last slide, and instead invested our $9,500 elsewhere at 5 percent, our FV would be less than the $10,000 the investment promised, and we would be worse off in FV terms : $9,500×(1.05) = $9,975 < $10,000 McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. 4.2 The Multiperiod Case The general formula for the future value of an investment over many periods can be written as: FV = C0×(1 + r)T Where C0 is cash flow at date 0, r is the appropriate interest rate, and T is the number of periods over which the cash is invested. McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Future Value Suppose a stock currently pays a dividend of $1.10, which is expected to grow at 40% per year for the next five years. What will the dividend be in five years? FV = C0×(1 + r)T $5.92 = $1.10×(1.40)5 McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Future Value and Compounding Notice that the dividend in year five, $5.92, is considerably higher than the sum of the original dividend plus five increases of 40percent on the original $1.10 dividend: $5.92 > $1.10 + 5×[$1.10×.40] = $3.30 This is due to compounding. McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Future Value and Compounding $1.10 (1.40) $1.10 (1.40) $1.10 (1.40) $1.10 (1.40) 5 4 3 2 $1.10 (1.40) $1.10 $1.54 $2.16 $3.02 $4.23 $5.92 0 1 2 3 4 5 McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Present Value and Discounting How much would an investor have to set aside today in order to have $20,000 five years from now if the current rate is 15%? PV $20,000 0 1 $9,943.53 2 4 5 $20,000 (1.15) McGraw-Hill/Irwin 3 5 Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. How Long is the Wait? If we deposit $5,000 today in an account paying 10%, how long does it take to grow to $10,000? FV C0 (1 r ) $10,000 $5,000 (1.10) T (1.10) T $10,000 T 2 $5,000 ln( 1.10) ln( 2) T T ln( 2) ln( 1.10) McGraw-Hill/Irwin 0.6931 7.27 years 0.0953 Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. What Rate Is Enough? Assume the total cost of a college education will be $50,000 when your child enters college in 12 years. You have $5,000 to invest today. What rate of interest must you earn on your investment to cover the cost of your child’s education? About 21.15%. FV C0 (1 r ) (1 r ) 12 T $50,000 $5,000 r 10 1 12 McGraw-Hill/Irwin $50,000 $5,000 (1 r ) 12 10 (1 r ) 10 1 12 1 1.2115 1 .2115 Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Calculator Keys Texas Instruments BA-II Plus FV = future value PV = present value I/Y = periodic interest rate P/Y must equal 1 for the I/Y to be the periodic rate Interest is entered as a percent, not a decimal N = number of periods Remember to clear the registers (CLR TVM) after each problem Other calculators are similar in format McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Multiple Cash Flows Consider an investment that pays $200 one year from now, with cash flows increasing by $200 per year through year 4. If the interest rate is 12%, what is the present value of this stream of cash flows? If the issuer offers this investment for $1,500, should you purchase it? McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Multiple Cash Flows 0 1 200 2 3 4 400 600 800 178.57 318.88 427.07 508.41 1,432.93 McGraw-Hill/Irwin Present Value < Cost → Do Not Purchase Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Valuing “Lumpy” Cash Flows First, set your calculator to 1 payment per year. Then, use the cash flow menu: CF0 0 CF3 600 I CF1 200 F3 1 NPV F1 1 CF4 800 CF2 400 F4 1 F2 1 McGraw-Hill/Irwin 12 1,432.93 Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. 4.3 Compounding Periods Compounding an investment m times a year for T years provides for future value of wealth: r FV C0 1 m McGraw-Hill/Irwin mT Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Compounding Periods For example, if you invest $50 for 3 years at 12% compounded semi-annually, your investment will grow to .12 FV $50 1 2 McGraw-Hill/Irwin 23 $50 (1.06) $70.93 6 Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Effective Annual Rates of Interest A reasonable question to ask in the above example is “what is the effective annual rate of interest on that investment?” FV $50 (1 .12 ) 23 $50 (1.06) $70.93 6 2 The Effective Annual Rate (EAR) of interest is the annual rate that would give us the same end-of-investment wealth after 3 years: $50 (1 EAR) $70.93 3 McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Effective Annual Rates of Interest FV $50 (1 EAR) $70.93 3 (1 EAR) 3 $70.93 $50 13 $70.93 EAR $50 1 .1236 So, investing at 12.36% compounded annually is the same as investing at 12% compounded semi-annually. McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Effective Annual Rates of Interest Find the Effective Annual Rate (EAR) of an 18% APR loan that is compounded monthly. What we have is a loan with a monthly interest rate rate of 1½%. This is equivalent to a loan with an annual interest rate of 19.56%. r 1 m McGraw-Hill/Irwin nm 12 .18 1 12 (1.015) 12 1.1956 Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. EAR on a Financial Calculator Texas Instruments BAII Plus keys: description: [2nd] [ICONV] Opens interest rate conversion menu Sets 12 payments per year [↑] [C/Y=] 12 [ENTER] Sets 18 APR. [↓][NOM=] 18 [ENTER] [↓] [EFF=] [CPT] McGraw-Hill/Irwin 19.56 Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Continuous Compounding The general formula for the future value of an investment compounded continuously over many periods can be written as: FV = C0×erT Where C0 is cash flow at date 0, r is the stated annual interest rate, T is the number of years, and e is a transcendental number approximately equal to 2.718. ex is a key on your calculator. McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. 4.4 Simplifications Perpetuity Growing perpetuity A stream of cash flows that grows at a constant rate forever Annuity A constant stream of cash flows that lasts forever A stream of constant cash flows that lasts for a fixed number of periods Growing annuity McGraw-Hill/Irwin A stream of cash flows that grows at a constant rate for a fixed number of periods Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Perpetuity A constant stream of cash flows that lasts forever 0 PV PV C C C 1 2 3 C C C (1 r ) (1 r ) 2 (1 r ) … 3 C r McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Perpetuity: Example What is the value of a British consol that promises to pay £15 every year for ever? The interest rate is 10-percent. 0 £15 £15 £15 1 2 3 PV £15 … £150 .10 McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Growing Perpetuity A growing stream of cash flows that lasts forever 0 PV PV McGraw-Hill/Irwin C C×(1+g) C ×(1+g)2 1 2 3 C (1 r ) C (1 g ) (1 r ) 2 C (1 g ) (1 r ) 3 … 2 C rg Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Growing Perpetuity: Example The expected dividend next year is $1.30, and dividends are expected to grow at 5% forever. If the discount rate is 10%, what is the value of this promised dividend stream? $1.30 $1.30×(1.05) $1.30 ×(1.05)2 … 0 1 PV McGraw-Hill/Irwin 2 $1.30 .10 .05 3 $26.00 Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Annuity A constant stream of cash flows with a fixed maturity C C C C 0 PV 1 2 3 C C C (1 r ) (1 r ) 2 (1 r ) T 3 C (1 r ) T C 1 PV 1 T r (1 r ) McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Annuity: Example If you can afford a $400 monthly car payment, how much car can you afford if interest rates are 7% on 36month loans? $400 $400 $400 $400 0 1 2 3 36 $400 1 PV $12,954.59 1 36 .07 / 12 (1 .07 12) McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. What is the present value of a four-year annuity of $100 per year that makes its first payment two years from today if the discount rate is 9%? 4 PV1 t 1 $297.22 $100 (1.09) $323.97 0 t $100 1 (1.09) $100 1 2 $327 .97 $297 .22 PV McGraw-Hill/Irwin 0 1.09 $100 (1.09) 2 $100 3 $100 (1.09) 3 $100 (1.09) $100 4 4 $323.97 $100 5 Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Growing Annuity A growing stream of cash flows with a fixed maturity C C×(1+g) C ×(1+g)2 C×(1+g)T-1 0 PV 1 C (1 r ) 2 3 C (1 g ) (1 r ) 2 T 1 g C PV 1 r g (1 r ) McGraw-Hill/Irwin T C (1 g ) (1 r ) T 1 T Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Growing Annuity: Example A defined-benefit retirement plan offers to pay $20,000 per year for 40 years and increase the annual payment by three-percent each year. What is the present value at retirement if the discount rate is 10 percent? $20,000 $20,000×(1.03) $20,000×(1.03)39 0 1 2 40 40 $20,000 1.03 PV $265,121.57 1 .10 .03 1.10 McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Growing Annuity: Example You are evaluating an income generating property. Net rent is received at the end of each year. The first year's rent is expected to be $8,500, and rent is expected to increase 7% each year. What is the present value of the estimated income stream over the first 5 years if the discount rate is 12%? $8,500 (1.07) $8,500 (1.07) 3 $8,500 (1.07) $8,500 (1.07) 2 $8,500 0 1 $34,706.26 McGraw-Hill/Irwin $9,095 $9,731.65 2 3 4 $10,412.87 $11,141.77 4 5 Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. 4.5 What Is a Firm Worth? Conceptually, a firm should be worth the present value of the firm’s cash flows. The tricky part is determining the size, timing and risk of those cash flows. McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Quick Quiz How is the future value of a single cash flow computed? How is the present value of a series of cash flows computed. What is the Net Present Value of an investment? What is an EAR, and how is it computed? What is a perpetuity? An annuity? McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved.